The Finiteness Conjecture for the Joint Spectral Radius of a Pair of Matrices

Abstract

A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. We study the finite-step realizability of the joint/generalized spectral radius of a pair of n × n square matrices. Let Σ = {A, B} where A,B are n × n matrices and B is a rank-one matrix. Then we have ρ(Σ)= max:t,s ρ(A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> B <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/(s+t)</sup> . That is to say, Σ have the finiteness property where the maximum is attained at (t, s) with the optimal sequence A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> B <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> .